Constraints on the richness-mass relation and the optical-SZE positional offset distribution for SZE-selected clusters

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(1)Publication Year. 2015. Acceptance in OA@INAF. 2020-09-16T11:33:26Z. Title. Constraints on the richness-mass relation and the optical-SZE positional offset distribution for SZE-selected clusters. Authors. A. Saro; S. Bocquet; E. Rozo; B. A. Benson; J. Mohr; et al.. DOI. 10.1093/mnras/stv2141. Handle. http://hdl.handle.net/20.500.12386/27415. Journal. MONTHLY NOTICES OF THE ROYAL ASTRONOMICAL SOCIETY. Number. 454.

(2) MNRAS 454, 2305–2319 (2015). doi:10.1093/mnras/stv2141. Constraints on the richness–mass relation and the optical-SZE positional offset distribution for SZE-selected clusters. Affiliations are listed at the end of the paper Accepted 2015 September 14. Received 2015 September 9; in original form 2015 June 25. ABSTRACT. We cross-match galaxy cluster candidates selected via their Sunyaev–Zel’dovich effect (SZE) signatures in 129.1 deg2 of the South Pole Telescope 2500d SPT-SZ survey with optically identified clusters selected from the Dark Energy Survey science verification data. We identify 25 clusters between 0.1 z 0.8 in the union of the SPT-SZ and redMaPPer (RM) samples. RM is an optical cluster finding algorithm that also returns a richness estimate for each cluster. We model the richness λ-mass relation with the following function ln λ|M500 ∝ Bλ ln M500 + Cλ ln E(z) and use SPT-SZ cluster masses and RM richnesses λ to constrain the parameters. +0.77 We find Bλ = 1.14+0.21 −0.18 and Cλ = 0.73−0.75 . The associated scatter in mass at fixed richness is. . E-mail: saro@usm.lmu.de. C 2015 The Authors Published by Oxford University Press on behalf of the Royal Astronomical Society. Downloaded from https://academic.oup.com/mnras/article/454/3/2305/1201112 by guest on 14 September 2020. A. Saro,1,2‹ S. Bocquet,1,2 E. Rozo,3 B. A. Benson,4,5,6 J. Mohr,1,2,7 E. S. Rykoff,8,9 M. Soares-Santos,4 L. Bleem,5,10 S. Dodelson,4,5,6 P. Melchior,11,12 F. Sobreira,4,13 V. Upadhyay,14 J. Weller,2,7,15 T. Abbott,16 F. B. Abdalla,17 S. Allam,4 R. Armstrong,18 M. Banerji,19 A. H. Bauer,20 M. Bayliss,21,22 A. Benoit-Lévy,17 G. M. Bernstein,18 E. Bertin,23 M. Brodwin,24 D. Brooks,17 E. Buckley-Geer,4 D. L. Burke,8,9 J. E. Carlstrom,5,6,25 R. Capasso,1,2 D. Capozzi,26 A. Carnero Rosell,13,27 M. Carrasco Kind,28,29 I. Chiu,1,2 R. Covarrubias,29 T. M. Crawford,5,6 M. Crocce,20 C. B. D’Andrea,26 L. N. da Costa,13,27 D. L. DePoy,30 S. Desai,1 T. de Haan,31,32 H. T. Diehl,4 J. P. Dietrich,1,2 P. Doel,17 C. E Cunha,8 T. F. Eifler,18,33 A. E. Evrard,34,35 A. Fausti Neto,13 E. Fernandez,36 B. Flaugher,4 P. Fosalba,20 J. Frieman,4,5 C. Gangkofner,1,2 E. Gaztanaga,20 D. Gerdes,35 D. Gruen,7,15 R. A. Gruendl,28,29 N. Gupta,1,2 C. Hennig,1,2 W. L. Holzapfel,32 K. Honscheid,11,12 B. Jain,18 D. James,16 K. Kuehn,37 N. Kuropatkin,4 O. Lahav,17 T. S. Li,30 H. Lin,4 M. A. G. Maia,13,27 M. March,18 J. L. Marshall,30 Paul Martini,11,38 M. McDonald,39 C.J. Miller,34,35 R. Miquel,36 B. Nord,4 R. Ogando,13,27 A. A. Plazas,33,40 C. L. Reichardt,32,41 A. K. Romer,42 A. Roodman,8,9 M. Sako,18 E. Sanchez,43 M. Schubnell,35 I. Sevilla,28,43 R. C. Smith,16 B. Stalder,22,44 A. A. Stark,22 V. Strazzullo,1,2 E. Suchyta,11,12 M. E. C. Swanson,29 G. Tarle,35 J. Thaler,45 D. Thomas,26 D. Tucker,4 V. Vikram,10 A. von der Linden,8,46,47 A. R. Walker,16 R. H. Wechsler,8,9,46 W. Wester,4 A. Zenteno48 and K. E. Ziegler5.

(3) 2306. A. Saro et al. σln M|λ = 0.18+0.08 −0.05 at a characteristic richness λ = 70. We demonstrate that our model provides an adequate description of the matched sample, showing that the fraction of SPT-SZ-selected clusters with RM counterparts is consistent with expectations and that the fraction of RMselected clusters with SPT-SZ counterparts is in mild tension with expectation. We model the optical-SZE cluster positional offset distribution with the sum of two Gaussians, showing that it is consistent with a dominant, centrally peaked population and a subdominant population characterized by larger offsets. We also cross-match the RM catalogue with SPT-SZ candidates below the official catalogue threshold significance ξ = 4.5, using the RM catalogue to provide optical confirmation and redshifts for 15 additional clusters with ξ ∈ [4, 4.5].. 1 I N T RO D U C T I O N Clusters of galaxies were first identified as overdense regions in the projected number counts of galaxies (e.g. Abell 1958; Zwicky, Herzog & Wild 1968). Nowadays, clusters are also regularly identified through their X-ray emission (e.g. Gioia et al. 1990; Vikhlinin ˇ et al. 1998; Böhringer et al. 2000; Pacaud et al. 2007; Suhada et al. 2012), and at millimetre wavelengths through their Sunyaev– Zel’dovich effect (SZE) signatures (Sunyaev & Zel’dovich 1972). Large, homogeneously selected samples of clusters are useful for both cosmological and astrophysical studies, and such samples have recently begun to be produced using SZE selection (Staniszewski et al. 2009; Hasselfield et al. 2013; Bleem et al. 2015b; Planck Collaboration 2015). There is a longer history of large cluster samples selected from optical and near-infrared photometric surveys (e.g. Gladders & Yee 2000; Koester et al. 2007; Eisenhardt et al. 2008; Hao et al. 2010; Menanteau et al. 2010; Wen, Han & Liu 2012; Ascaso, Wittman & Dawson 2014; Bleem et al. 2015a; Rykoff et al. 2014, and references therein), and even larger samples will soon be available from ongoing and future surveys like the Dark Energy Survey (DES; The Dark Energy Survey Collaboration 2005),1 KiDS (de Jong et al. 2013), Euclid (Laureijs et al. 2011) and LSST (LSST Dark Energy Science Collaboration 2012). Reliable estimates of galaxy cluster masses play a key role in both cosmological and astrophysical cluster studies. First, the abundance of galaxy clusters as a function of mass is a well-known cosmological probe (White, Efstathiou & Frenk 1993; Bartlett & Silk 1994; Eke et al. 1998; Viana & Liddle 1999; Borgani et al. 2001; Vikhlinin et al. 2009; Mantz et al. 2010; Rozo et al. 2010; Allen, Evrard & Mantz 2011; Benson et al. 2013; Bocquet et al. 2015; Mantz et al. 2015, and many others). Secondly, accurate estimates of cluster masses are crucial in disentangling environmental effects from the secular evolution processes shaping galaxy formation (Mei et al. 2009; Zenteno et al. 2011; Muzzin et al. 2012). In this paper, we calibrate the richness–mass relation for SZEselected galaxy clusters detected in the DES science verification data (SVA1) using the red-sequence Matched-Filter Probabilistic Percolation(redMaPPer; Rykoff et al. 2014) cluster-finding algorithm. Specifically, we study the clusters detected via their SZE signatures in the South Pole Telescope SPT-SZ cluster survey (Bleem et al. 2015b, hereafter B15) that are also present in the redMaPPer catalogue. We also study the distribution of offsets between the SZE-derived centres and the associated optical centres, properly including the SZE positional uncertainties. Finally, we demonstrate 1. http://www.darkenergysurvey.org. MNRAS 454, 2305–2319 (2015). our ability to push to even lower candidate significance within the SPT-SZ candidate catalogue by taking advantage of the contiguous, deep, multiband imaging available through DES. In this respect, our study points towards the combined use of DES and SPT data sets to provide highly reliable extended SZE-selected cluster samples. We note that historically the optical follow-up of SPT-selected clusters was the original motivation for proposing DES. The plan of the paper is as follows. In Section 2, we describe the galaxy cluster catalogues and the matching metric we use in this work. Section 3 describes the method we adopt to calibrate the SZE–mass and richness–mass relations. Our results are presented in Section 4. Section 5 contains a discussion of our findings and our conclusions. In the appendix, we provide a preliminary analysis of a cluster sample created using an independent cluster finding algorithm – the Voronoi Tessellation (VT) cluster finder – which helps to highlight areas where the VT algorithm can be improved. Throughout this work, we adopt M = 0.3, = 0.7, H0 = 70 km s−1 Mpc−1 and σ 8 = 0.8. Cluster masses are defined within R500 , the radius within which the density is 500 times the critical density of the Universe. Future analyses will include the dependence of the derived scaling-relation parameters on the adopted cosmology by simultaneously fitting for cosmological and scaling-relation parameters (e.g. Mantz et al. 2010; Rozo et al. 2010; Bocquet et al. 2015).. 2 CLUSTER SAMPLE 2.1 SPT-SZ cluster catalogue The SPT-SZ galaxy cluster sample used in this analysis has been selected via the cluster thermal SZE signatures in a point-source masked-region of 2365 deg2 of the 2540 deg2 (2500d) SPT-SZ survey using 95 and 150 GHz data. Typical instrumental noise is approximately 40 (18) μKCMB arcmin and the beam FWHM is 1.6 (1.2) arcmin for the 95 (150) GHz maps. A multi-frequency matched filter is used to extract the cluster SZE signal in a manner designed to optimally measure the cluster signal given knowledge of the cluster profile, the SZE spectrum and the noise in the maps (Haehnelt & Tegmark 1996; Melin, Bartlett & Delabrouille 2006). The cluster gas profiles are assumed to be described by a projected isothermal β model (Cavaliere & Fusco-Femiano 1976) with β = 1. Note that, as discussed in Vanderlinde et al. (2010), the resulting SPT-SZ candidate catalogues are not sensitive to this assumption. The adopted model provides an SZE temperature decrement that is. Downloaded from https://academic.oup.com/mnras/article/454/3/2305/1201112 by guest on 14 September 2020. Key words: catalogues – methods: data analysis – galaxies: abundances – galaxies: clusters: general – galaxies: haloes – galaxies: statistics – cosmology: miscellaneous – large-scale structure of Universe..

(4) λ-mass and optical-SZE offset distribution maximum at the cluster centre and weakens with separation θ from the cluster centre as. T (θ) = T0 [1 + (θ/θc )2 ]−1 ,. (1). where T0 is the central value and θ c is the core radius. We adopt 12 different cluster profiles linearly spaced from θ c = 0.25 to 3 arcmin (Vanderlinde et al. 2010; Reichardt et al. 2013, B15). For each cluster, the maximum signal to noise across the 12 filtered maps is denoted as ξ . The SPT-SZ cluster candidates with ξ > 4.5 have been previously published in B15.. The DES Science Verification Data (DES-SVA1) that overlap SPT have been used to produce optically selected catalogues of clusters. In Section 2.2.1, we describe the acquisition and preparation of the DES-SVA1 data, and in Section 2.2.2, we describe the production of the redMaPPer cluster catalogue used in the primary analysis. We remind the reader that in Appendix A we present results of a preliminary analysis of the VT cluster catalogue.. 2.2.1 DES-SVA1 data The DES-SVA1 data include imaging of ∼300 deg2 over multiple disconnected fields (Sánchez et al. 2014; Banerji et al. 2015; Melchior et al. 2015), most of which overlap with the SPT-SZ survey. The DES-SVA1 data were acquired with the Dark Energy Camera (Diehl et al. 2012; Flaugher et al. 2012, 2015) over 78 nights, starting in Fall 2012 and ending early in 2013 with depth comparable to the nominal depth of the full DES survey (Rykoff et al., in preparation). Data have been processed through the DES Data Management (DESDM; Desai et al. 2012) pipeline that is an advanced version of development versions described in several publications (Ngeow et al. 2006; Mohr et al. 2008, 2012). The data were calibrated in several stages leading to a Gold catalogue of DES-SVA1 galaxies (Rykoff et al., in preparation). The Gold catalogue covers ∼250 deg2 and is optimized for extragalactic science. In particular, it masks regions south of declination δ = −61◦ , avoiding the Large Magellanic Cloud and its high stellar densities. Furthermore, the footprint is restricted to the regions where we have coverage in all four bands.. probability of every galaxy in the vicinity of a galaxy cluster. The richness λ is thus defined as the sum of the membership probabilities (pRM ) over all galaxies (see section 4 of Rykoff et al. 2014 for more details): λ= pRM . (2) In addition to the estimate of membership probabilities, the RM centring algorithm is also probabilistic. The centring probability Pcen is a likelihood-based estimate of the probability that the galaxy under consideration is a central galaxy. The centring likelihood includes the fact that the photometric redshift of the central galaxy must be consistent with the cluster redshift, that the central galaxy luminosity must be consistent with the expected luminosity of the central galaxy of a cluster of the observed richness, and that that the galaxy density on a 300 kpc scale consistent with the galaxy density of central galaxies. The centring probability further accounts for the fact that every cluster has one and only one central galaxy, properly accounting for the relevant combinatoric factors. These probabilities have been tested on SDSS DR8 data using X-ray-selected galaxy clusters, and have been shown to produce cluster centres that are consistent with the X-ray centres (Rozo & Rykoff 2014). The DES-SVA1 RM catalog was produced by running on a smaller footprint than that for the full SVA1 Gold sample. In particular, we restrict the catalogue to the regions where the z-band 10σ galaxy limiting magnitude is z > 22. In total, we use 148 deg2 of DES-SVA1 imaging, with 129.1 deg2 overlapping the SPT-SZ footprint. In this area, the largest fraction (124.6 deg2 ) is included in the so called DES-SVA1 SPT-E field. The final catalogue used in this work consists of 9281 clusters with λ > 5 and redshifts in the range 0.1 < z < 0.9. Due to the varying depth of the DES-SVA1 catalog, RM produces a mask that determines the maximum redshift of the cluster search at any given location in the survey. As an example, the effective area in the SPT-E region at the highest redshift (z > 0.85) is only ∼30 deg2 . In addition to the cluster catalogue, the RM algorithm also uses the survey mask to produce a set of random points with the same richness and redshift distribution as the clusters in the catalogue. The random points take into account the survey geometry and the physical extent of the clusters, and as with the clusters, only includes points that have <20 per cent of the local region masked (see Rykoff et al. 2014). 2.3 Catalogue matching. 2.2.2 redMaPPer cluster catalogue The red-sequence Matched-Filter Probabilistic Percolation (redMaPPer; hereafter RM) algorithm is a cluster-finding algorithm based on the richness estimator of Rykoff et al. (2012). RM has been applied to photometric data from the Eighth Data Release (DR8) of the Sloan Digital Sky Survey (SDSS; Aihara et al. 2011) and to the SDSS Stripe 82 co-add data (Annis et al. 2014), and has been shown to provide excellent photometric redshifts, richness estimates that tightly correlate with external mass proxies, and very good completeness and purity (Rozo & Rykoff 2014; Rozo et al. 2014a, 2015b). We refer the reader to the paper by Rykoff et al. (2014) for a detailed description of the algorithm. Here, we briefly summarize the most salient features. We employ an updated version of the algorithm (v6.3.3), with improvements summarized in Rozo & Rykoff (2014), Rozo et al. (2015a) and Rykoff et al. (in preparation). RM calibrates the colour of red-sequence galaxies using galaxy clusters with spectroscopic redshifts. RM uses this information to estimate the membership. We cross-match the SPT-SZ catalogue with the RM optical cluster catalogue following the method of Rozo et al. (2015b). First, we sort the SPT-SZ clusters to produce a list with decreasing SZE observable ξ , and we sort the RM catalogue to produce a list with decreasing richness. Secondly, we go down the SPT-SZ sorted list, associating each SPT cluster candidate with the richest RM cluster candidate whose centre lies within 1.5 R500 of the SZE centre. Thirdly, we remove the associated RM cluster from the list of possible counterparts when matching the remaining SPT-selected clusters. R500 is first computed assuming the redshift of the optical counterpart and using the SZE-mass scaling-relation parameters adopted in B15. We subsequently check that our sample does not change when adopting our best-fitting scaling-relation parameters (see Section 3.1). To test the robustness of our matching algorithm against chance associations, we first perform the above described procedure on a sample of randomly generated RM clusters as described in the previous section. Positions of clusters in this randomly generated sample MNRAS 454, 2305–2319 (2015). Downloaded from https://academic.oup.com/mnras/article/454/3/2305/1201112 by guest on 14 September 2020. 2.2 DES optical cluster catalogues. 2307.

(5) 2308. A. Saro et al.. do not correlate with the positions of the SPT-SZ clusters. Using an ensemble of 104 random catalogues, we measure the distribution of richness in chance associations for each SPT-SZ candidate. We then apply the algorithm to match the real RM cluster catalogue with the SPT-SZ candidate list where ξ > 4.5. The distribution of probabilities p of chance associations estimated for each cluster candidate using the randomly generated samples is shown in Fig. 1 (filled histogram). In Fig. 2 (left-hand panel) we show the resulting 84 and 95 per cent confidence limits (solid and dotted lines, respectively) in the richness distribution of the chance associations as a function of the SPT-SZ observable ξ . This test allows us to estimate the probability of chance superposition for each SPT-SZ cluster candidate. As detailed below, we use this information to determine whether or not to include particular matches for further analysis. As Fig. 2 shows, this filtering of the matched sample then ensures that chance superpositions are playing no more than a minor role even at 4 < ξ < 4.5. Within the DES-SVA1 region explored by RM there are 36 such SPT-SZ cluster candidates. Using information on the contamination fraction at ξ > 4.5 of the SPT-SZ candidate list and on the redshift distribution of the confirmed cluster candidates (Song et al. 2012, B15), we expect approximately nine of these candidates to be noise fluctuations and ∼80 per cent of the real clusters to lie at z < 0.8. Therefore, assuming the optical catalogue is complete, the expected number of real cluster matches is ∼22. Similarly, one can estimate an expected number of real cluster matches (23.6) by scaling the total number of confirmed clusters in B15 below z = 0.8 (433) in the 2365 deg2 of the SPT-SZ survey by the DES-SVA1 area overlapping SPT-SZ that has been processed with the RM cluster finder (129.1 deg2 ). The actual number of matches to the SPT-SZ candidates is 33. Eight systems are foreground, low-richness RM clusters that have been erroneously associated with SPT-SZ candidates with either previously measured redshifts (four systems) or lower limits estimated in B15 (the remaining four candidates) that are at z 0.8; these systems are either noise fluctuations or real clusters that are at redshifts too high for them to be detected by RM. In fact, all of these systems have a probability p of chance associations estimated MNRAS 454, 2305–2319 (2015). Downloaded from https://academic.oup.com/mnras/article/454/3/2305/1201112 by guest on 14 September 2020. Figure 1. Distribution of probability p of chance associations for the ξ > 4.5 (filled) and ξ > 4 (open) samples. Black lines show the sample used in this work, red lines show the sample rejected (Section 2.3).. from the randomly generated sample that is p > 16 per cent. Therefore, we remove these matches from the sample (red histograms in Fig. 1). This leaves 25 SPT-SZ candidates at ξ > 4.5 that have RM counterparts (black histograms in Fig. 1). We expect less than one false associations within this sample of 25 candidates. The associated optical richness as a function of the SPT-SZ significance is shown in the left-hand panel of Fig. 2. This number is somewhat larger but statistically consistent with the expected number of matches presented above. All 22 of the SPT-SZ confirmed clusters presented in B15 that lie at redshifts where they could be detected by RM are in this matched sample. According to our matching metric (which differs from the approach in B15), there are also three unconfirmed SPT-SZ candidates (i.e. candidates without identified optical counterparts in the B15 analysis) that have RM counterparts: SPTCL J0502-6048, SPT-CL J0437-5307 and SPT-CL J0500-4551. The newly confirmed clusters are highlighted with large circles in Fig. 2. Of the 25 SPT-SZ candidates with robust RM counterparts, we use 19 of them to calibrate the RM richness–mass relation. Six clusters are excluded from the analysis for the following reasons. Clusters with estimated redshift z < 0.25 in the SPT-SZ catalogue from B15 are highlighted in cyan in the left-hand panel of Fig. 2. Because the ξ –mass relation is robust only above this redshift (Vanderlinde et al. 2010), these systems are not used in the following analysis. Two systems (SPT-CL J0440-4744 and SPT-CL J0441-4502) are excluded from this analysis as they are detected in SPT-SZ regions that have been masked due to their proximity to point sources, which can compromise the SZE signal-to-noise measurement. In addition, we exclude the three clusters highlighted in magenta: SPT-CL J0417-4748, SPT-CL J0456-5116 and SPT-CL J0502-6048. These systems are strongly masked in the DES-SVA1 data; based on the SZE position, the masks cover 40 per cent of the total cluster region. As a result, the associated optical counterparts are highly miscentred, and the corresponding richness is severely biased. We note that the average centring failure rate caused by the detection mask is 12 per cent (3 clusters out of 25), in comparison to the corresponding rate in the SDSS RM catalogue, which is ≈1– 2 per cent. The difference reflects the fact that SDSS has a much larger contiguous area, while SVA1 has a more aggressive star mask. We expect this failure rate will decrease as the DES coverage increases, and object masking improves. Furthermore, improvements will be made to the RM algorithm to estimate the masked area not only at the putative centre of the cluster, but at all possible centres. In this way, clusters at high risk of mask-induced miscentring will be properly removed from the sample. The B15 catalogue contains only SPT-SZ candidates with ξ ≥ 4.5. In this work, we also apply the matching algorithm to SPT-SZ candidates at 4 < ξ < 4.5. We identify 26 matches in this signal-tonoise range. The resulting probabilities of chance associations of the ξ > 4 sample are shown as open histograms in Fig. 1. Similarly to the ξ > 4.5 case, we exclude 11 of these systems, which have estimated probabilities p of chance associations p > 16 per cent. For the 15 matched systems, the expected number of false associations is also smaller than one. The remaining cleaned sample is shown as red points on the left-hand panel of Fig. 2. The resulting total number of SPT-SZ and RM associations at ξ > 4 is 40. This number is in good agreement with the expectation (∼36) obtained using the number of SPT candidates above ξ > 4 in the DES-SVA1 region explored by RM (88) and correcting it by the expected number of noise fluctuations (∼45) and the number of clusters above z > 0.8 (∼7). We find that two ξ < 4.5 SPT-SZ candidates, SPT-CL J0501-4717 and SPT-CL J0439-5611, have probabilities of random associations.

(6) λ-mass and optical-SZE offset distribution. 2309. larger than 5 per cent, and therefore it is not clear whether these low-richness associations are correct (see Fig. 2). The right-hand panel of Fig. 2 contains a comparison of the redshifts from the RM catalogues with the redshifts published in B15 (zSPT ) for the same clusters (obtained through dedicated optical/NIR followup by the SPT team or taken from the literature). Clusters with spectroscopic redshifts are highlighted in green. We note that the redshift estimates are not biased for the clusters affected by masking (magenta points). For SPT-SZ candidates with 4 < ξ < 4.5 the SPT collaboration did not complete followup optical imaging, and therefore we adopt the redshifts of the RM optical cluster counterpart. Table 1 contains all SPT candidates with RM counterparts used in this work. For newly confirmed SPT-SZ clusters, the associated zSPT redshift is not given. We caution that the masses for low-redshift clusters (z < 0.25) may be underestimated due to filtering that is done to remove the noise component associated with the primary CMB.. with the RM cluster catalogue cleans the SPT-SZ candidate list, removing the expected noise fluctuations. Within this context the resulting cluster sample is therefore drawn from the halo mass function through the SPT-SZ selection in the redshift range explored by the RM catalogue. In the following subsections, we describe the model we use to simultaneously constrain the SZE–mass relation (Section 3.1) and the richness–mass relation (Section 3.2).. 3 M A S S C A L I B R AT I O N M E T H O D. where the bias in ξ is due to maximizing the signal to noise over three variables (cluster right ascension, declination and core radius). The scatter of the actual observable ξ with respect to ξ is characterized by a Gaussian of unit width. The SPT observable– mass relation P(ζ |M500 , z) is modelled as a lognormal distribution of mean M500 lnζ |M500 , z = lnASZE + BSZE ln 3 × 1014 h−1 M

(7) E(z) + CSZE ln (4) E(z = 0.6). We apply the method described in Bocquet et al. (2015) to characterize the λ–mass relation of SPT-selected clusters. We refer the reader to the original paper (and to the appendix in particular) for a detailed description of the method. A similar approach has been adopted by Liu et al. (2015) for studying the SZE properties of an X-ray-selected cluster sample from the XMM-BCS survey (Desai ˇ et al. 2012; Suhada et al. 2012). In this analysis, we consider the RM richness as a follow-up observable to the SZE-selected cluster sample. This choice is adequate as there are no SPT-SZ candidates with ξ > 4.5 missing RM counterparts in the redshift and spatial regime explored by the RM catalogue, so that the cross-sample can indeed be thought of as solely SPT selected. We note that this is not the case for SPT-SZ candidates with 4 < ξ ≤ 4.5 that do not have RM counterparts. However, the adopted method is also accurate under the assumption that cross-matching the SPT-SZ candidate list. 3.1 The SZE–mass relation Following previous SPT papers (Vanderlinde et al. 2010; Benson et al. 2013; Reichardt et al. 2013; Bocquet et al. 2015, B15), we define the unbiased SZE significance ζ as the average signal to noise a cluster would produce over many realizations of SPT data, if the cluster position and core radius were perfectly known. This quantity is related to the expectation value of ξ over many realizations of the SPT data by (3) ζ = ξ 2 − 3,. and scatter DSZE , and where E(z) ≡ H(z)/H0 . At low significance ζ 2, there is a non-negligible chance of multiple low-mass clusters overlapping within the same resolution element of the SPT beam. We account for this by only considering the brightest of these objects per approximate resolution element and we compute P(ζ max |ζ ) MNRAS 454, 2305–2319 (2015). Downloaded from https://academic.oup.com/mnras/article/454/3/2305/1201112 by guest on 14 September 2020. Figure 2. Left-hand panel: richness as a function of the SZE significance ξ for the matched cluster sample. SPT-SZ candidates with ξ < 4.5 (vertical line) are shown in red. Clusters at z < 0.25 (cyan) and clusters with miscentering due to a high masked fraction (magenta) are not used in the richness analysis. Large circles indicate the newly confirmed SPT-SZ candidates with ξ > 4.5. Solid and dashed lines represent the upper 84 and 95 percentiles in richness of chance associations of SPT-SZ candidates and clusters from the randomly generated RM catalogue. Right-hand panel: the estimated redshift for the RM sample as a function of SPT redshifts as presented in B15 from independent optical follow-up data. SPT-SZ candidates with spectroscopic redshift are shown in green. Magenta symbols are the same as in the left-hand panel..

(8) 2310. A. Saro et al.. Table 1. SPT-SZ cluster candidates with RM counterpart. We report the SPT-ID (1), right ascension (2) and declination (3), SPT peak detection significance ξ (4), corresponding core radius (5), richness λ (6), associated redshift from the RM catalogue (7) and SPT catalogue (8), and SPT derived masses (9). Coordinates are J2000. SPT ID. Dec.. ξ. θ c (arcmin). λ. zRM. zSPT. M500 [1014 h−1 70 M

(9) ]. 69.574 10.199 64.344 79.149 72.273 74.115 70.450 69.807 70.229 71.843 65.591 69.928 68.249 83.791 70.242 67.026 71.162 74.598 83.606 75.724 70.345 67.430 73.002 69.259 75.209 73.307 72.473 65.809 72.937 69.564 74.099 67.970 75.274 79.507 69.655 78.273 72.888 69.978 83.237 72.472. −54.319 −44.133 −47.812 −54.510 −49.023 −51.275 −48.917 −46.012 −46.964 −50.921 −51.674 −53.502 −56.502 −59.939 −47.736 −60.828 −43.872 −57.695 −59.625 −60.810 −45.040 −52.559 −48.108 −53.119 −45.856 −50.451 −44.672 −55.104 −50.965 −46.488 −45.523 −53.896 −47.294 −57.670 −49.117 −59.029 −49.178 −56.192 −57.877 −59.142. 22.88 19.34 14.24 12.41 8.91 8.58 8.56 8.28 7.13 5.97 5.86 5.61 5.32 5.20 5.12 5.11 5.01 4.87 4.74 4.69 4.62 4.56 4.52 4.51 4.51 4.47 4.37 4.36 4.34 4.31 4.30 4.22 4.20 4.19 4.19 4.17 4.14 4.14 4.11 4.11. 0.50 0.50 0.25 1.50 0.50 1.00 0.50 0.25 1.25 0.25 1.00 0.75 1.75 0.25 1.25 1.25 1.50 2.50 0.25 0.25 2.50 0.75 0.50 0.25 0.75 0.25 0.75 1.25 0.50 0.50 0.25 0.50 3.00 0.25 1.75 0.25 0.25 0.50 0.50 1.25. 144.76 ± 5.52 137.45 ± 7.03 54.22 ± 6.75 178.93 ± 8.71 91.37 ± 4.75 73.08 ± 5.39 86.96 ± 4.55 55.18 ± 3.52 67.95 ± 3.62 77.84 ± 5.26 49.28 ± 5.32 60.77 ± 3.81 60.75 ± 4.82 50.25 ± 4.09 82.55 ± 3.80 55.91 ± 5.95 70.53 ± 5.91 37.90 ± 2.68 40.43 ± 3.42 30.73 ± 4.32 51.22 ± 4.11 33.84 ± 3.97 56.54 ± 4.89 36.89 ± 3.56 32.68 ± 4.82 31.99 ± 3.44 54.50 ± 5.43 38.65 ± 3.44 83.11 ± 4.62 41.01 ± 3.37 32.45 ± 3.18 57.23 ± 4.76 19.47 ± 2.50 65.22 ± 4.82 76.20 ± 4.16 33.29 ± 3.58 54.10 ± 4.10 17.82 ± 2.93 48.49 ± 3.81 107.14 ± 5.29. 0.42 ± 0.01 0.37 ± 0.01 0.58 ± 0.01 0.33 ± 0.02 0.80 ± 0.01 0.56 ± 0.01 0.81 ± 0.01 0.34 ± 0.02 0.33 ± 0.02 0.40 ± 0.01 0.58 ± 0.01 0.43 ± 0.01 0.71 ± 0.02 0.67 ± 0.02 0.30 ± 0.02 0.73 ± 0.02 0.57 ± 0.01 0.19 ± 0.01 0.58 ± 0.01 0.79 ± 0.02 0.15 ± 0.01 0.52 ± 0.01 0.42 ± 0.01 0.29 ± 0.02 0.26 ± 0.01 0.77 ± 0.02 0.15 ± 0.00 0.27 ± 0.02 0.76 ± 0.01 0.43 ± 0.01 0.29 ± 0.02 0.75 ± 0.02 0.35 ± 0.02 0.82 ± 0.01 0.24 ± 0.01 0.61 ± 0.01 0.73 ± 0.02 0.28 ± 0.02 0.77 ± 0.02 0.77 ± 0.01. 0.42 0.35 0.58 0.29 0.79 0.56 0.79 ± 0.04 0.34 ± 0.04 0.35 ± 0.04 0.39 ± 0.05 0.59 ± 0.03 0.43 ± 0.04 0.69 0.58 ± 0.03 − 0.64 ± 0.03 0.57 ± 0.03 0.19 ± 0.02 0.58 − − 0.53 ± 0.03 0.37 ± 0.04 − − − − − − − − − − − − − − − − −. 10.19 ± 1.33 9.71 ± 1.28 7.41 ± 1.00 7.05 ± 0.97 5.24 ± 0.78 5.39 ± 0.81 5.10 ± 0.77 5.52 ± 0.84 4.95 ± 0.81 4.24 ± 0.81 3.98 ± 0.78 3.97 ± 0.81 3.56 ± 0.78 3.46 ± 0.77 3.75 ± 0.83 3.46 ± 0.79 3.53 ± 0.82 3.69 ± 0.85 3.15 ± 0.76 3.03 ± 0.76 3.49 ± 0.85 3.15 ± 0.79 3.26 ± 0.81 3.20 ± 0.80 3.66 ± 0.91 2.89 ± 0.74 3.42 ± 0.86 3.26 ± 0.83 2.81 ± 0.74 3.07 ± 0.79 3.17 ± 0.81 2.74 ± 0.73 3.34 ± 0.88 2.60 ± 0.70 3.13 ± 0.81 2.75 ± 0.73 2.71 ± 0.73 3.10 ± 0.81 2.59 ± 0.71 2.68 ± 0.73. following Crawford et al. (2010). The SPT observable–mass relation is therefore expanded to P(ζ max |ζ )P(ζ |M500 , z) and ζ max is then converted to the observable ξ as in equation (3). To calibrate the ζ –M relation we use the subsample of clusters with ξ > 5 and z > 0.25 from the 2500d SPT-SZ catalogue (B15). We determine the parameter values by abundance-matching the catalogue against our fixed reference cosmology. We predict the expected number of clusters as a function of mass and redshift using the halo mass function (Tinker et al. 2008). We convolve this mass function with the observable–mass relation accounting for its associated uncertainties, and compare the prediction with the data. Our approach here is effectively the opposite of the typical analysis, where cosmological parameters are deduced from the cluster sample using both priors and calibrating information to constrain the scaling-relation parameters (e.g. Benson et al. 2013; Bocquet et al. 2015); here, we assume perfect knowledge of cosmology to calibrate the scaling relation. Note that this method does not depend on any assumptions about hydrostatic equilibrium.. MNRAS 454, 2305–2319 (2015). We assume flat priors on ASZE , BSZE , CSZE and a Gaussian prior on DSZE = 0.18 ± 0.07; the latter corresponds to the posterior distribution derived from the cosmological analysis of the full SPT sample (de Haan et al., in preparation). We obtain the following parameters for the ζ –mass relation by maximizing the likelihood of obtaining the observed sample in ξ and redshift under the model derived from equations (3) and (4). The results are ASZE = 4.02 ± 0.16, BSZE = 1.71 ± 0.09, CSZE = 0.49 ± 0.16, DSZE = 0.20 ± 0.07.. (5). For every cluster in the sample we also calculate the associated mass distribution, accounting for selection effects: P (M500 |ξ, z, p) ∝ P (ξ |M500 , z, p) P (M500 |z, p),. (6). where the vector p encapsulates cosmological and scaling-relation parameters and P (ξ |M500 , z, p) is obtained from the ξ –mass scaling relation as described by equations (3) and (4). The halo mass. Downloaded from https://academic.oup.com/mnras/article/454/3/2305/1201112 by guest on 14 September 2020. SPT-CL J0438-5419 SPT-CL J0040-4407 SPT-CL J0417-4748 SPT-CL J0516-5430 SPT-CL J0449-4901 SPT-CL J0456-5116 SPT-CL J0441-4855 SPT-CL J0439-4600 SPT-CL J0440-4657 SPT-CL J0447-5055 SPT-CL J0422-5140 SPT-CL J0439-5330 SPT-CL J0433-5630 SPT-CL J0535-5956 SPT-CL J0440-4744 SPT-CL J0428-6049 SPT-CL J0444-4352 SPT-CL J0458-5741 SPT-CL J0534-5937 SPT-CL J0502-6048 SPT-CL J0441-4502 SPT-CL J0429-5233 SPT-CL J0452-4806 SPT-CL J0437-5307 SPT-CL J0500-4551 SPT-CL J0453-5027 SPT-CL J0449-4440 SPT-CL J0423-55062 SPT-CL J0451-5057 SPT-CL J0438-4629 SPT-CL J0456-4531 SPT-CL J0431-5353 SPT-CL J0501-4717 SPT-CL J0518-5740 SPT-CL J0438-4907 SPT-CL J0513-5901 SPT-CL J0451-4910 SPT-CL J0439-5611 SPT-CL J0532-5752 SPT-CL J0449-5908. RA.

(10) λ-mass and optical-SZE offset distribution function P (M500 |z, p) is the prior on the mass distribution at redshift z. Masses derived for the matched cluster sample are shown in Table 1. We note that both these masses and the SZE scaling-relation parameters quoted here are different from the ones reported in B15. We adopt the same fixed cosmology as in B15, but in this analysis we consider data from the full SPT-SZ cluster-survey as opposed to just the sample from the initial 720 deg2 (Reichardt et al. 2013).. 2311. Table 2. Best-fitting parameters and 68 per cent confidence level of the richness–mass scaling-relation parameters described by equations (7) and (8). Catalogue SPT-SZ+RM ξ > 4.5 SPT-SZ+RM ξ > 4. Aλ. Bλ. Cλ. Dλ. 66.1+6.3 −5.9 69.8+6.0 −4.9. 1.14+0.21 −0.18 1.17+0.19 −0.17. 0.73+0.77 −0.75 1.71+0.63 −0.57. 0.15+0.10 −0.07 0.20+0.09 −0.08. 3.2 Richness–mass relation. where Aλ is the normalization, Bλ characterizes the mass dependence and Cλ characterizes the redshift evolution. An additional parameter Dλ describes the intrinsic scatter in λ, which is assumed to be lognormal and uncorrelated with the SZE scatter, with variance given by Var(lnλ|M500 ) = exp(−lnλ|M500 ) + Dλ2 .. (8). The first term above represents the Poisson noise associated with the number of galaxies in a halo at fixed mass, and therefore we define the intrinsic scatter Dλ as lognormal scatter in addition to Poisson noise. We assume flat priors on the distributions of Aλ , Bλ , Cλ and a positive flat prior for Dλ . The probability that a cluster with SPT-SZ signal-to-noise ξ is observed to have a richness λ is (9) P (λ|ξ, z, p) = dM500 P (λ|M500 , z, p)P (M500 |ξ, z, p). The term P (λ|M500 , z, p) contains the lognormal intrinsic scatter and normal measurement uncertainties in the observable λ. We use the above distribution to evaluate the likelihood of the matched cluster sample defined through our cross-matching procedure. Note that we simultaneously vary both the optical and SZE scaling-relation parameters, further including the SZE data set from B15 with ξ > 5, z > 0.25 for constraining the SZE–mass relation. 4 R E S U LT S We present here the constraints on the richness–mass relation (Section 4.1) and then use these best-fitting parameters to explore whether the cumulative distribution of the matched samples are consistent with the expectations from the model (Section 4.2). Finally, in Section 4.3, we analysis the optical-SZE positional offset distribution. 4.1 redMaPPer richness–mass relation We marginalize over the SZE-mass scaling-relation parameters and constrain the posterior distributions for the RM λ–mass scaling 2 SPT-CL J0423-5506 was previously identified in Song et al. (2012) and Reichardt et al. (2013) at a redshift z = 0.21 ± 0.04 with a signal to noise ξ = 4.51. The associated redshift estimated with the same analysis presented in B15 is z = 0.25 ± 0.036.. Figure 3. Posterior distribution for the four parameters of the λ–mass scaling relation (equations 7 and 8). Predictions from the scaling relation of Rykoff et al. (2012) are shown as dashed black lines. Best-fitting parameters and associated 1σ uncertainties appear in Table 2.. relation. Our best-fitting parameters and 68 per cent confidence level intervals are reported in Table 2 and shown in Fig. 3. We note that the slope of the λ–mass relation is consistent with 1 within 1σ (consistent, therefore, with the richness being proportional to the mass), and the model is consistent with no redshift evolution within 1σ (Andreon & Congdon 2014). Furthermore the resulting λ–mass relation is characterized by a remarkably low asymptotic intrinsic scatter, with σlnλ|M500 → 0.15+0.10 −0.07 as lnλ|M500 → ∞ (equation 8). Following Evrard et al. (2014), we estimate the characteristic scatter in mass at fixed richness to arrive at σln M = 0.18+0.08 −0.05 at λ = 70, ∼25 per cent larger than the corresponding characteristic scatter in mass at fixed ξ . We present in Fig. 4 the RM richness as a function of the SPT derived masses. Blue lines describe the best-fitting model and intrinsic scatter as derived from this analysis (Table 2) at a pivot point of z = 0.6. We have verified that our results are not dominated by uncertainties in the SZE–mass scaling relation by fixing these parameters to their best-fitting values. Our results are only marginally improved in this case. Consequently, future analyses with larger samples are expected to considerably reduce the uncertainties of the recovered λ–mass scaling-relation parameters. In Fig. 4, one cluster appears to be an obvious outlier: SPT-CL J0516-5435 (ξ = 12.4, λ = 178.9, M500 = 7.05×1014 h−1 70 M

(11) ). SPT-CL J0516-5345 is a well-known merger that is elongated in a north–south direction in the plane of the sky with an X-ray mass estimate nearly a factor of two larger than the SZE mass estimate. MNRAS 454, 2305–2319 (2015). Downloaded from https://academic.oup.com/mnras/article/454/3/2305/1201112 by guest on 14 September 2020. As for the SZE–mass relation (equation 4), we assume a power-law form for the λ–mass relation: M500 lnλ|M500 , z = lnAλ + Bλ ln 3 × 1014 h−1 M

(12) E(z) + Cλ ln , (7) E(z = 0.6).

(13) 2312. A. Saro et al.. 4.2 Consistency test of model Figure 4. Richness as a function of the SPT derived masses for the calibration sample used in this analysis (Section 2.3). Blue lines show the best-fitting richness–mass relation and 1σ intrinsic scatter.. This cluster was in fact the strongest outlier in the sample of 14 clusters in Andersson et al. (2011). High et al. (2012) made a weak lensing measurement of SPT-CL J0516-5345, and found that there was a significant offset between the brightest central galaxy (BCG) and the weak-lensing centre, consistent with the merger hypothesis. Additionally, High et al. (2012) found that the weak-lensing mass was in better agreement with the SZE-mass estimate than the X-ray mass estimate, at a level consistent with elongation observed in the plane of the sky. Therefore, SPT-CL J0516-5345 appears to be an outlier due to true intrinsic scatter in the observable–mass relations, so we leave it in our analysis. We note, however, that whether or not we include SPT-CL J0516-5345 in the fit has a significant impact on our results. Our best-fitting parameters shift from Bλ = +0.10 +0.17 1.14+0.21 −0.18 and Dλ = 0.15−0.07 with this cluster, to Bλ = 1.00−0.15 and Dλ = 0.05+0.07 when SPT-CL J0516-5345 is not included in −0.03 the fit. Whether SPT-CL J0516-5345 represents a rare event in a non-Gaussian tail in the distribution of richness of galaxy clusters or the recovered lognormal scatter obtained when cluster SPT-CL J0516-5349 is included is more correct will thus need to await future analyses with larger samples. We convert the P(M500 |λ) scaling relation derived by Rykoff et al. (2012) using abundance matching and the SDSS maxBCG clustercatalogue (Koester et al. 2007) to a richness–mass relation so that we can compare it to our results (see also Evrard et al. 2014). The predictions from Rykoff et al. (2012) are shown as dashed lines in Fig. 3 under the assumption of no redshift evolution in the richness–mass relation. We note that all parameters of our derived RM λ–mass scaling relation for SPT-selected clusters are consistent with the Rykoff et al. (2012) values. We repeat these analyses extending the sample to include those with 4 < ξ < 4.5 and find similar results (Table 2). The largest difference is in the redshift evolution term which now has a bestfitting value Cλ = 1.71+0.63 −0.57 . While formally this difference does not have large statistical significance (1.3σ ), it is coming from a sample that includes a large fraction of the same clusters, so it is likely statistically significant. A larger redshift evolution term would imply that higher redshift RM clusters are less massive at fixed λ. At the same time the derived scatter is also larger. MNRAS 454, 2305–2319 (2015). We also test the consistency of the adopted scaling relation model with the data by examining whether we are finding the expected number of matches with the correct distribution in richness. To do this, we focus on the SPT-E field, which at ∼124.6 deg2 is the largest contiguous region covered by the DES-SVA1 data. We carry out two different tests. In the first test, we examine whether we are finding the expected number of SZE-selected clusters and whether these clusters have the expected number of optical matches with the correct λ distribution. We generate 106 Monte Carlo realizations of cluster samples extracted from the halo mass function above M500 > 1013.5 h−1 70 M

(14) and assign richnesses λ and SPT-SZ significance ξ using the parameters we extract from our analysis of the real matched catalogue. These Monte Carlo mocks are generated taking into account the survey area as a function of redshift sampled by RM in the SPT-E field. We then apply the SZE selection – either ξ > 4.5 or ξ > 4 – and measure the cumulative distribution in λ of the SZE-selected samples. We then compare this to the same distribution in the real matched catalogue. Shaded regions in the upper panels of Fig. 5 show 1σ , 2σ and 3σ confidence regions obtained from the mocks after marginalizing over the scaling-relation parameters. The solid black line shows the distribution from the real catalogue, which is in good agreement with the mocks. The largest observed difference is smaller than 1σ indicating that the adopted model provides a consistent description of the observed number and richness distribution of the SZE-selected sample. Similarly, a Kolmogorov–Smirnov (KS) test for the observed cumulative distribution of matched systems as a function of λ and the corresponding median distribution from the Monte Carlo simulations, shows that the null hypothesis of data being drawn from same distributions cannot be excluded and returns p −values 0.76 and 0.96 for ξ > 4.5 and ξ > 4, respectively. The second test is focused on whether the RM cluster catalogue (which is significantly larger than the SPT-SZ catalogue) has the expected number of SZE matches with the correct λ distribution. Essentially, we take the observed RM catalogue as a starting point, and calculate the expected number of systems with SPT-SZ counterparts given the model and parameter constraints from our λ-mass likelihood analysis. This test differs from the first in that it takes the observed RM-selected cluster sample as a starting point, and such a test should be more sensitive to, for example, contamination in the RM catalogue. The formally correct way to evaluate a statistical difference between the RM-selected sample and the. Downloaded from https://academic.oup.com/mnras/article/454/3/2305/1201112 by guest on 14 September 2020. Two of the clusters in the 4 < ξ < 4.5 range are compatible with false associations. Excluding the two matched clusters with the highest probability of random associations results in an ∼1σ shift in Bλ (from Bλ = 1.17 to Bλ = 1.04) and in an ∼0.5σ shift in Cλ (from Cλ = 1.71 to Cλ = 1.42), while the other parameters (Aλ and Dλ ) are almost unchanged. Interestingly, even though we have increased the number of clusters in the SPT-SZ+RM sample by 40 per cent, the constraints on the scaling-relation parameters are only mildly tighter. There are two reasons for this. First these lower signal-to-noise SPT-SZ clusters have larger fractional mass uncertainties in ξ (ξ −1 ∼ 0.23 and ξ −1 ∼ 0.13, respectively for the 4 < ξ < 4.5 and ξ > 4.5 samples). Secondly, the richnesses are also systematically lower, leading to a larger Poisson variance. Thus, each low-ξ cluster has less constraining power than a high-ξ cluster, reducing the impact of extending the sample to include the lower mass systems..

(15) λ-mass and optical-SZE offset distribution. 2313. SPT-SZ+RM matched sample used in this analysis would be to calibrate the λ–mass and SZE–mass relations starting from the RMselected sample. Such a study goes, however, beyond the scope of the current work and will be addressed in a future project. Therefore, the following analysis is only intended to be a consistency check. For this purpose, we proceed by first computing, for each real RM-selected cluster in the SPT-E field, the probability Pm of that cluster also having ξ > 4.5 and therefore being in the matched sample. We define this probability as ∞ P (ξ |λ, z)dξ Pm = 4.5 ∞ = dξ dM500 P (ξ |M500 )P (M500 |λ, z), (10) 4.5. where P(M500 |λ, z) ∝ P(λ|M500 , z)P(M500 , z) and P(M500 , z) is the halo mass function (Tinker et al. 2008). To predict the expected number of RM clusters with SZE counterparts, we then randomly sample the scaling-relation parameters, determining Pm for all the RM clusters in each case. We then Monte Carlo sample each Pm to produce randomly sampled matched cluster catalogues. We use the results from the ensemble. of random matched catalogues to produce the expected cumulative distributions in λ. Orange, yellow and red regions in the bottom panels of in Fig. 5 show the 1σ , 2σ and 3σ confidence regions, respectively, of the expected cumulative distribution in λ for ξ > 4.5 (left-hand) and ξ > 4 (right-hand), given the RM catalogue and scaling-relation parameter constraints as input. The dashed blue line shows the cumulative distribution of the entire sample of RM clusters in the area, while the solid black line shows the observed cumulative distribution of the real matched catalogue. We note that the predicted number of SPT-SZ+RM matches in this case tends to be higher than that observed for λ > 35, but the tension is weak. Of some concern is the high λ end of the sample (λ > 70), where only 9 of the 17 RM selected clusters have SPT-SZ counterparts at ξ > 4.5 despite their having large probabilities indicating they should be in the SZE-selected sample. However, a KS test for the observed cumulative distribution of matched systems as a function of λ and the corresponding median distribution from the mocks shows that the null hypothesis of data being drawn from the same parent distribution cannot be excluded and returns p −values 0.90 and 0.33, respectively, for ξ > 4.5 and ξ > 4. While the KS test is showing no evidence for the two distributions to differ, it is also known to be not very sensitive to the tails of the. MNRAS 454, 2305–2319 (2015). Downloaded from https://academic.oup.com/mnras/article/454/3/2305/1201112 by guest on 14 September 2020. Figure 5. Consistency tests of our model (Section 3) and the SPT-SZ+RM catalogue from the ∼124 deg2 SPT-E field. The solid black lines show the observed cumulative distribution in richness λ of the ξ > 4.5 (left-hand) and ξ > 4 (right-hand) SZE-selected samples. Upper panels: grey-scale regions show 1, 2 and 3σ regions predicted by drawing 106 SZE-selected samples from the mass function and assigning λ according to our scaling relation constraints. There is good agreement with the data. Lower panels: dashed blue lines show the cumulative distribution in λ of the full RM sample. Orange, yellow and red areas define 1, 2 and 3σ regions representing the predicted cumulative distribution of the SPT-SZ+RM catalogue using as input the full RM sample and the probability (equation 10) that each RM cluster will have an SPT-SZ counterpart, given our scaling relation constraints..

(16) 2314. A. Saro et al.. distributions (Moscovich-Eiger, Nadler & Spiegelman 2013). With this respect, it is also therefore interesting to directly compare the observed cumulative distribution with the cumulative distribution from the predicted number of matches. In this case, as for the classical KS test, we focus on the largest difference between the two distributions. Lower-left panel of Fig. 5 shows that at most, the observed cumulative distribution is in tension with expectations at the ∼2σ level, providing some indication of tension between the observed sample and our model. For example, when restricting ourselves to clusters with λ > 70, we find that our simulation results in a larger number of SPT+RM associations in 94.6 per cent of our Monte Carlo realizations. This tendency to observe fewer matches than expected given the size of the RM-selected sample could be explained by either contamination within the RM sample, additional incompleteness within the SPT-SZ sample beyond that caused by scatter in the SZE-mass relation, or simply a statistical fluctuation. Future work exploring the SZE properties of the lower mass systems along with an extension of the current analysis to the full overlap between DES and the SPT-SZ survey will sharpen this test. 4.3 Optical-SZE positional offset distribution It has been shown that optical-miscentering can have a significant impact on the derived SZE signature for an optically selected sample (Biesiadzinski et al. 2012; Sehgal et al. 2013; Rozo & Rykoff 2014; Rozo et al. 2014a,b). We note, however, that in our case the SZE signal has been estimated at the SZE-determined position, so our results are not affected by optical miscentering. In fact, we can now use our data to constrain the distribution of offsets between the SZE-determined and the optically determined cluster centres. As an example, we show in Fig. 6 the DES-SVA1 gri pseudo colour image of SPT-CL J0433-5630, an SPT-SZ selected cluster with ξ ∼ 5.3 at redshift z = 0.69 (B15). Yellow contours show the SPT-SZ signal to noise in steps of ξ = 1, while the magenta circle describes the projected radius R500 /2. The cyan label refers to the associated RM cluster centre. This RM cluster has richness λ ∼ 60. We note that the most probable central galaxy selected by RM is significantly offset from the SZE defined centre. As a result, the MNRAS 454, 2305–2319 (2015). Figure 7. Filled histogram shows the measured fraction of SPT-SZ+RM clusters as a function of the optical-SZE positional offset in units of R500 . The green curve shows the SPT-SZ positional uncertainty, and the blue curves shows the best-fitting SZE-optical positional offset model.. measured SZE signature at the optical position (ξ = 4.1) would be significantly underestimated with respect to the derived unbiased quantity ζ = 5 obtained through equation (3). We stress that this effect is not important for the scaling relation results reported in Section 4.1, as the sample analysed here is SZE selected. Fig. 7 contains a normalized histogram of the distribution of cluster positional offsets in units of R500 for the ξ > 4.5 analysed SPT-SZ sample. Under the assumption that the measurement uncertainty from the optical side is negligible, we model this distribution as an underlying intrinsic positional offset distribution convolved with the SPT-SZ positional uncertainty. The 1σ SPT-SZ positional uncertainty for a cluster with a pressure profile given by a spherical β model with β = 1 and core radius θ c , detected with SPT-SZ significance ξ is described by 2 + θc2 , (11). θ = ξ −1 θbeam where θ beam = 1.2 arcmin is the beam FWHM (see Story et al. 2011 and equation 4 in Song et al. 2012 for more details). As a result, the expected distribution of positional offsets in the case in which the intrinsic one is a δ-function is shown (arbitrarily rescaled) as a green line. Song et al. (2012) have shown that the intrinsic optical-SZE positional offset distribution for an SPT-SZ selected sample is consistent with the optical-X-ray positional offset distribution of X-ray selected clusters (Lin, Mohr & Stanford 2004). The offset distribution can be characterized by a large population of central galaxies with small offsets from the SZE centres and a less populated tail of central galaxies with large offsets (e.g. Lin et al. 2004; Rozo & Rykoff 2014; Lauer et al. 2014). We therefore parametrize the distribution of positional offsets between the RM centre and the SZE centre for x as. 2 2 1 − ρ0 − 2σx 2 ρ0 − 2σx 2 e 0 + e 1 (12) P (x) = 2πx 2πσ02 2πσ12 where x = r/R500 , ρ 0 is the fraction of the population with small offsets and is characterized by variance σ02 , while the population with larger offsets is characterized by variance σ12 . While this model for the distribution was motivated by the expected intrinsic positional. Downloaded from https://academic.oup.com/mnras/article/454/3/2305/1201112 by guest on 14 September 2020. Figure 6. SPT-CL J0433-5630: DES-SVA1 gri pseudo colour image overplotted with SPT-SZ signal-to-noise contours (in steps ξ = 1). The magenta circle shows the projected R500 /2 radius at z = 0.69, while the green circle describes the 1σ SPT positional uncertainty (equation 11). The cyan label marks the associated λ ∼ 60 RM cluster..

(17) λ-mass and optical-SZE offset distribution Table 3. Best-fitting parameters and 68 per cent confidence level of the optical-SZE positional offset distribution. Catalogue RM-ξ > 4.5. ρ0 0.63+0.15 −0.25. σ 0 [R500 ] 0.07+0.03 −0.02. σ 1 [R500 ] 0.25+0.07 −0.06. offset distribution, the measured distribution will include both the actual physical SZE-central galaxy offset distribution and the systematics due to failures in identifying the correct cluster centre with the RM algorithm. For every cluster and parameter ρ 0 ∈ [0, 1], σ 0 ∈ [0, 1], and σ 1 ∈ [σ 0 , 1], we then compare the predicted offset distribution obtained by convolving the model with the SPT-SZ positional uncertainty of equation (11) to extract the associated likelihood. Best-fitting parameters and 68 per cent confidence intervals are shown in Table 3 and joint and fully marginalized parameter constraints are shown in Fig. 8. We note that the positional offset distribution for the RM sample is consistent with a concentrated dominant population (ρ0 = +0.03 0.63+0.15 −0.25 ) of smaller offsets (σ0 = 0.07−0.02 R500 ) and a subdominant population characterized by larger offsets (σ1 = 0.25+0.07 −0.06 R500 ). In the limit where the SZ centre and BCG are coincident for every cluster, the RM code(Rykoff et al. 2012, 2014) predicts the SZ-RM offset distribution for this sample to contain a centrally peaked component with normalization Pcen to be 0.79. While this is formally consistant with the measured SZ - RM offset distribution, intrinsic optical-SZE positional offsets, such as those presented by (e.g. Song et al. 2012) likely contribute to large separation component of the observed distribution shown in Fig. 7. Larger samples will be necessary to disentangle the impact of systematics due to failures in identifying the correct cluster centre with the RM algorithm from the offset distribution due to cluster morphology.. 5 CONCLUSIONS In this paper, we cross-match SZE-selected cluster candidates with ξ > 4.5 from the 2500d SPT-SZ survey (B15) with the optically selected cluster catalogue extracted from the DES science verification data DES-SVA1. The optically selected catalogue is created using the RM cluster-finding algorithm. We study the robustness of our matching algorithm by applying it to randomly generated RM catalogues. Using the adopted matching algorithm in the 129.1 deg2 of overlap between the two data sets, we create a matched catalogue of 33 clusters. Eight of these clusters are removed as likely chance superpositions that are identified using the randomly generated catalogues. The resulting 25 cluster sample includes all previously known z < 0.8 and ξ > 4.5 SPT-SZ clusters in this area (Song et al. 2012; Reichardt et al. 2013, B15) in addition to three previously unconfirmed SPT-SZ clusters. We then study three characteristics of this cross-matched SPTSZ+RM cluster sample. (1) The richness–mass relation of SPT-SZ selected clusters. We calibrate the λ–mass relation from SZE measurements by applying the method described in Bocquet et al. (2015). In this analysis we assume a fixed fiducial cosmology and marginalize over the simultaneously calibrated SPT-SZ ξ -mass relation. We adopt flat priors on the richness–mass relation parameters. We find that the RM λ–mass relation for SPT-SZ selected clusters is characterized by a small asymptotic intrinsic scatter Dλ = 0.15+0.10 −0.07 and by a slope Bλ = 1.14+0.21 −0.18 that is consistent with unity. Our constraints are in good agreement with those of Rykoff et al. (2012) and show that the scatter in mass at fixed richness λ = 70 for this sample is only 25 per cent larger than the scatter in mass at fixed SPT-SZ observable ξ . (2) Consistency test of model and matched catalogue. We carry out two consistency tests to determine whether there is tension between the observed matched sample and the expectations given the scaling relation model we have adopted. Both tests involve creating Monte Carlo generated cluster catalogues with associated richness and SPT-SZ significance derived from the fitted scaling relations. The first test checks whether the correct number of SZE-selected clusters is found and whether those clusters exhibit the correct number of optical matches with the expected λ distribution. As is clear from Fig. 5, the observations are perfectly consistent with the expectations from the model. Thus, our analysis shows that the data in our matched SPT-SZ+RM sample are well described by our adopted model. In the second test, we take the much larger observed RM catalogue as a starting point and use the model to test whether the expected number of SZE matches with the expected λ distribution is found. Unlike the first test, this one would in principle be sensitive to contamination within the RM sample. Here, the agreement is not as good because there is a tendency for there to be fewer observed matches than expected. However, the tension reaches the ∼2σ level at worst, and so there is no convincing evidence that our observed sample is inconsistent with the model. (3) The SZE-optical positional offset distribution. We identify optical positional biases associated with 12 per cent of the sample due to the masking in the DES-SVA1 data. We remove these clusters and study the optical-SZE positional offset distribution for the rest of the matched sample. We model the underlying positional offset distribution as the sum of two Gaussians, while accounting for the SPT-SZ positional uncertainty. We show that the resulting distribution is consistent with being described by a dominant +0.03 (63+15 −25 per cent) centrally peaked distribution with (σ0 = 0.07−0.02 MNRAS 454, 2305–2319 (2015). Downloaded from https://academic.oup.com/mnras/article/454/3/2305/1201112 by guest on 14 September 2020. Figure 8. Posterior distribution for the 1σ and 2σ level of the three parameter model describing the positional offset distribution of equation (12). Best-fitting parameters are shown in Table 3.. 2315.

(18) 2316. A. Saro et al.. R500 ) and a sub-dominant (∼37 per cent) population characterized by larger separations (σ1 = 0.25+0.07 −0.06 R500 ). For the same population, the RM algorithm assumes that 79 per cent of the clusters will belong to a small-offset population, consistent with our observations.. AC K N OW L E D G E M E N T S We acknowledge the support by the DFG Cluster of Excellence ‘Origin and Structure of the Universe’, the Transregio program TR33 ‘The Dark Universe’ and the Ludwig-Maximilians University. The South Pole Telescope is supported by the National Science Foundation through grant PLR-1248097. Partial support is also provided by the NSF Physics Frontier Center grant PHY1125897 to the Kavli Institute of Cosmological Physics at the University of Chicago, the Kavli Foundation and the Gordon and Betty Moore Foundation grant GBMF 947. AAS acknowledges a Pell grant from the Smithsonian Institution. TDH is supported by a Miller Research Fellowship. This work was partially completed at Fermilab, operated by Fermi Research Alliance, LLC under contract no. De-AC02-07CH11359 with the United States Department of Energy. The Dark Cosmology Centre is funded by the Danish National Research Foundation. CR acknowledges support from the Australian Research Council’s Discovery Projects scheme (DP150103208). Funding for the DES Projects has been provided by the US Department of Energy, the US National Science Foundation, the Ministry of Science and Education of Spain, the Science and Technology Facilities Council of the United Kingdom, the Higher Education Funding Council for England, the National Center for Supercomputing Applications at the University of Illinois at Urbana-Champaign, the Kavli Institute of Cosmological Physics at the University of Chicago, the Center for Cosmology and AstroParticle Physics at the Ohio State University, the Mitchell Institute for Fundamental Physics and Astronomy at Texas A&M University, Financiadora de Estudos e Projetos, Fundaça˜ o Carlos Chagas Filho de Amparo a` Pesquisa do Estado do Rio de Janeiro, Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnológico and the Ministério da Ciência e Tecnologia, the Deutsche Forschungsgemeinschaft and the Collaborating Institutions in the Dark Energy Survey. 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Including the SPT-SZ candidates between ξ = 4 and 4.5 increases the sample of matched clusters by ∼40 per cent compared to the ξ > 4.5 sample, highlighting the potential synergies of SPT and DES in producing lower mass extensions of SZE-selected cluster samples. We show that this larger sample produces results that are broadly consistent with the ξ > 4.5 results, but only marginally tighter. This is due to the fact that mass constraints from lower signal-to-noise SPT clusters are somewhat weaker on a per cluster basis compared to the higher ξ sample. Future work benefiting from the larger region of overlap between the DES and SPT surveys will improve our derived constraints and help to better characterize the optical and SZE properties of cluster samples in terms of positional offsets, purity, and completeness. Moreover, the multiwavelength data sets available through DES and SPT enable characterization of the galaxy populations of large SZE-selected cluster samples, calibration of the SZE-selected cluster masses using weak lensing constraints, and many other promising studies.. ESP2013-48274, FPA2013-47986 and Centro de Excelencia Severo Ochoa SEV-2012-0234, some of which include ERDF funds from the European Union. The Collaborating Institutions are Argonne National Laboratory, the University of California at Santa Cruz, the University of Cambridge, Centro de Investigaciones Energeticas, Medioambientales y Tecnologicas-Madrid, the University of Chicago, University College London, the DES-Brazil Consortium, the Eidgenössische Technische Hochschule (ETH) Zürich, Fermi National Accelerator Laboratory, the University of Edinburgh, the University of Illinois at Urbana-Champaign, the Institut de Ciencies de l’Espai (IEEC/CSIC), the Institut de Fisica d’Altes Energies, Lawrence Berkeley National Laboratory, the Ludwig-Maximilians Universität and the associated Excellence Cluster Universe, the University of Michigan, the National Optical Astronomy Observatory, the University of Nottingham, The Ohio State University, the University of Pennsylvania, the University of Portsmouth, SLAC National Accelerator Laboratory, Stanford University, the University of Sussex and Texas A&M University..

(19) λ-mass and optical-SZE offset distribution. A P P E N D I X A : S T U DY O F M A S S - R I C H N E S S S C AT T E R F O R C L U S T E R S F O U N D W I T H T H E VT METHOD The scatter of the mass–richness relation is a primary source of systematic uncertainties in cosmological measurements using galaxy clusters. To ensure that this and other astrophysical systematics are kept under control in future cosmology analyses, the DES collaboration has pursued the development of multiple cluster-finding algorithms. Here, we present initial results obtained using the analysis framework described throughout this paper on a DES-SVA1. cluster catalogue created using the VT cluster finder (Soares-Santos et al. 2011). The VT method is fundamentally different from RM. Specifically, VT uses photometric redshifts to detect clusters in 2+1 dimensions, and is designed to produce a cluster catalogue up to z ∼ 1 and down to mass M ∼ 1013.5 M